September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is a important topic for anyone who wishes to master math or any related subject that employs it. It's hard work, but we'll ensure you get a grip of these theories so you can make the grade!

Don’t feel discouraged if you don’t recall or don’t have a good grasp on these theories, as this blog will teach you all the essentials. Furthermore, we will teach you the secret to learning faster and increasing your grades in mathematics and other popular subjects today.

The Theorem

The vertical angle theorem states that when two straight lines bisect, they create opposite angles, named vertical angles.

These opposite angles share a vertex. Furthermore, the most important point to bear in mind is that they also measure the same! This applies that regardless of where these straight lines cross, the angles opposite each other will constantly share the equal value. These angles are called congruent angles.

Vertically opposite angles are congruent, so if you have a value for one angle, then it is possible to discover the others employing proportions.

Proving the Theorem

Proving this theorem is somewhat easy. First, let's draw a line and call it line l. After that, we will draw another line that intersects line l at some point. We will call this second line m.

After drawing these two lines, we will label the angles created by the intersecting lines l and m. To prevent confusion, we labeled pairs of vertically opposite angles. Accordingly, we label angle A, angle B, angle C, and angle D as follows:

We know that angles A and B are vertically opposite because they share the same vertex but don’t share a side. Bear in mind that vertically opposite angles are also congruent, meaning that angle A is identical angle B.

If you observe angles B and C, you will note that they are not linked at their vertex but next to one another. They have in common a side and a vertex, therefore they are supplementary angles, so the sum of both angles will be 180 degrees. This situation repeats itself with angles A and C so that we can summarize this in the following manner:

∠B+∠C=180 and ∠A+∠C=180

Since both additions equal the same, we can sum up these operations as follows:


By removing C on both sides of the equation, we will end with:


So, we can say that vertically opposite angles are congruent, as they have the same measurement.

Vertically Opposite Angles

Now that we have studied about the theorem and how to prove it, let's talk particularly regarding vertically opposite angles.


As we mentioned, vertically opposite angles are two angles created by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.

Still, vertically opposite angles are no way next to each other. Adjacent angles are two angles that have a common side and a common vertex. Vertically opposite angles never share a side. When angles share a side, these adjacent angles could be complementary or supplementary.

In the case of complementary angles, the addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just utilized to prove the vertical angle theorem.

These concepts are relevant within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not meet the properties of vertically opposite angles.

There are various properties of vertically opposite angles. Still, chances are that you will only require these two to secure your exam.

  1. Vertically opposite angles are at all time congruent. Consequently, if angles A and B are vertically opposite, they will measure the same.

  2. Vertically opposite angles are never adjacent. They can share, at most, a vertex.

Where Can You Use Opposite Angles in Real-World Scenario?

You might wonder where you can utilize these concepts in the real world, and you'd be amazed to notice that vertically opposite angles are quite common! You can locate them in several daily objects and situations.

For example, vertically opposite angles are made when two straight lines cross. Inside your room, the door installed to the door frame makes vertically opposite angles with the wall.

Open a pair of scissors to make two intersecting lines and adjust the size of the angles. Road crossings are also a great example of vertically opposite angles.

In the end, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are created by the trunk and the branches.

Be sure to observe your environment, as you will find an example next to you.

Puttingit Together

So, to summarize what we have considered so far, vertically opposite angles are created from two intersecting lines. The two angles that are not adjacent have the same measure.

The vertical angle theorem explains that whenever two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tested by drawing a straight line and another line overlapping it and implementing the theorems of congruent angles to complete measures.

Congruent angles means two angles that measure the same.

When two angles share a side and a vertex, they can’t be vertically opposite. Despite that, they are complementary if the addition of these angles totals 90°. If the addition of both angles totals 180°, they are considered supplementary.

The total of adjacent angles is consistently 180°. Thus, if angles B and C are adjacent angles, they will at all time equal 180°.

Vertically opposite angles are pretty common! You can discover them in various everyday objects and circumstances, such as windows, doors, paintings, and trees.

Further Study

Search for a vertically opposite angles practice questions online for examples and exercises to practice. Math is not a onlooker sport; keep applying until these theorems are well-established in your brain.

Despite that, there is no shame if you require additional support. If you're struggling to grasp vertical angles (or any other ideas of geometry), consider signing up for a tutoring session with Grade Potential. One of our professional teachers can help you comprehend the topic and nail your following test.